The functions in this section give access to the action on the underlying root system (or datum) of a permutation Coxeter group. These functions do not apply to finitely presented Coxeter groups
In the following functions, the optional parameter Basis determines which basis the roots are given with respect to: "Standard" for the standard basis of the root space; "Root" for the basis of simple (co)roots; "Weight" for the basis of simple (co)weights.
Basis: MonStgElt Default: "Standard"
The G-set of the Coxeter group W acting on the (co)roots.
> W := CoxeterGroup("B3");
> X := RootGSet(W);
> r := Root(W, 5);
> r;
(0 1 1)
> Image(W.1, X, r);
(1 1 1)
Basis: MonStgElt Default: "Standard"
The map X x W to X giving the action of the Coxeter group W on the (co)root space X.
> W := CoxeterGroup("B3");
> act := CorootAction(W);
> act([1,-2,1], W.1);
(-1 -1 1)
Basis: MonStgElt Default: "Standard"
The Coxeter group W as a real reflection group (ie. as a matrix group over some subfield of R) acting on the (co)root space, and the isomorphism from W to the (co)reflection group.
> W := CoxeterGroup("B3");
> _, h := ReflectionGroup(W);
> W.1*W.3;
(1, 10)(2, 8)(3, 12)(4, 7)(5, 6)(11, 17)(13, 16)(14, 15)
> h(W.1*W.3);
[-1 0 0]
[ 1 1 2]
[ 0 0 -1]